3.22 \(\int \frac{A+B x+C x^2+D x^3}{(a+b x) (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=210 \[ -\frac{2 \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{3/2} (b c-a d)^{5/2}}+\frac{2 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (-A d^3-2 c^3 D+c^2 C d\right )\right )}{d^3 \sqrt{c+d x} (b c-a d)^2}+\frac{2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^3 (c+d x)^{3/2} (b c-a d)}+\frac{2 D \sqrt{c+d x}}{b d^3} \]

[Out]

(2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(3*d^3*(b*c - a*d)*(c + d*x)^(3/2)) + (2
*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(c^2*C*d - A*d^3 - 2*c^3*D)))/(d^3*(b*c -
a*d)^2*Sqrt[c + d*x]) + (2*D*Sqrt[c + d*x])/(b*d^3) - (2*(A*b^3 - a*(b^2*B - a*b
*C + a^2*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b^(3/2)*(b*c - a
*d)^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.633884, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094 \[ -\frac{2 \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{3/2} (b c-a d)^{5/2}}+\frac{2 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (-A d^3-2 c^3 D+c^2 C d\right )\right )}{d^3 \sqrt{c+d x} (b c-a d)^2}+\frac{2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^3 (c+d x)^{3/2} (b c-a d)}+\frac{2 D \sqrt{c+d x}}{b d^3} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x + C*x^2 + D*x^3)/((a + b*x)*(c + d*x)^(5/2)),x]

[Out]

(2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(3*d^3*(b*c - a*d)*(c + d*x)^(3/2)) + (2
*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(c^2*C*d - A*d^3 - 2*c^3*D)))/(d^3*(b*c -
a*d)^2*Sqrt[c + d*x]) + (2*D*Sqrt[c + d*x])/(b*d^3) - (2*(A*b^3 - a*(b^2*B - a*b
*C + a^2*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b^(3/2)*(b*c - a
*d)^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 156.355, size = 298, normalized size = 1.42 \[ \frac{2 D \sqrt{c + d x}}{b d^{3}} + \frac{2 \left (A b^{3} - B a b^{2} + C a^{2} b - D a^{3}\right )}{b^{2} \sqrt{c + d x} \left (a d - b c\right )^{2}} - \frac{2 \left (C b d - D a d - 2 D b c\right )}{b^{2} d^{3} \sqrt{c + d x}} - \frac{2 \left (A b^{3} - B a b^{2} + C a^{2} b - D a^{3}\right )}{3 b^{3} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{2 \left (B b^{2} d^{2} - C a b d^{2} - C b^{2} c d + D a^{2} d^{2} + D a b c d + D b^{2} c^{2}\right )}{3 b^{3} d^{3} \left (c + d x\right )^{\frac{3}{2}}} + \frac{2 \left (A b^{3} - B a b^{2} + C a^{2} b - D a^{3}\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{b^{\frac{3}{2}} \left (a d - b c\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((D*x**3+C*x**2+B*x+A)/(b*x+a)/(d*x+c)**(5/2),x)

[Out]

2*D*sqrt(c + d*x)/(b*d**3) + 2*(A*b**3 - B*a*b**2 + C*a**2*b - D*a**3)/(b**2*sqr
t(c + d*x)*(a*d - b*c)**2) - 2*(C*b*d - D*a*d - 2*D*b*c)/(b**2*d**3*sqrt(c + d*x
)) - 2*(A*b**3 - B*a*b**2 + C*a**2*b - D*a**3)/(3*b**3*(c + d*x)**(3/2)*(a*d - b
*c)) - 2*(B*b**2*d**2 - C*a*b*d**2 - C*b**2*c*d + D*a**2*d**2 + D*a*b*c*d + D*b*
*2*c**2)/(3*b**3*d**3*(c + d*x)**(3/2)) + 2*(A*b**3 - B*a*b**2 + C*a**2*b - D*a*
*3)*atan(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c))/(b**(3/2)*(a*d - b*c)**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 1.0851, size = 201, normalized size = 0.96 \[ \frac{2 \sqrt{c+d x} \left (\frac{3 b \left (A d^3+2 c^3 D-c^2 C d\right )-3 a d \left (B d^2+3 c^2 D-2 c C d\right )}{(c+d x) (b c-a d)^2}+\frac{A d^3-B c d^2+c^3 (-D)+c^2 C d}{(c+d x)^2 (b c-a d)}+\frac{3 D}{b}\right )}{3 d^3}-\frac{2 \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{3/2} (b c-a d)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x + C*x^2 + D*x^3)/((a + b*x)*(c + d*x)^(5/2)),x]

[Out]

(2*Sqrt[c + d*x]*((3*D)/b + (c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)/((b*c - a*d)*(c
+ d*x)^2) + (-3*a*d*(-2*c*C*d + B*d^2 + 3*c^2*D) + 3*b*(-(c^2*C*d) + A*d^3 + 2*c
^3*D))/((b*c - a*d)^2*(c + d*x))))/(3*d^3) - (2*(A*b^3 - a*(b^2*B - a*b*C + a^2*
D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b^(3/2)*(b*c - a*d)^(5/2)
)

_______________________________________________________________________________________

Maple [B]  time = 0.026, size = 464, normalized size = 2.2 \[ 2\,{\frac{D\sqrt{dx+c}}{b{d}^{3}}}-{\frac{2\,A}{3\,ad-3\,bc} \left ( dx+c \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,Bc}{3\, \left ( ad-bc \right ) d} \left ( dx+c \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,{c}^{2}C}{3\,{d}^{2} \left ( ad-bc \right ) } \left ( dx+c \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,D{c}^{3}}{3\,{d}^{3} \left ( ad-bc \right ) } \left ( dx+c \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{Ab}{ \left ( ad-bc \right ) ^{2}\sqrt{dx+c}}}-2\,{\frac{Ba}{ \left ( ad-bc \right ) ^{2}\sqrt{dx+c}}}+4\,{\frac{Cac}{d \left ( ad-bc \right ) ^{2}\sqrt{dx+c}}}-2\,{\frac{Cb{c}^{2}}{{d}^{2} \left ( ad-bc \right ) ^{2}\sqrt{dx+c}}}-6\,{\frac{Da{c}^{2}}{{d}^{2} \left ( ad-bc \right ) ^{2}\sqrt{dx+c}}}+4\,{\frac{Db{c}^{3}}{{d}^{3} \left ( ad-bc \right ) ^{2}\sqrt{dx+c}}}+2\,{\frac{A{b}^{2}}{ \left ( ad-bc \right ) ^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-2\,{\frac{bBa}{ \left ( ad-bc \right ) ^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+2\,{\frac{C{a}^{2}}{ \left ( ad-bc \right ) ^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-2\,{\frac{D{a}^{3}}{b \left ( ad-bc \right ) ^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((D*x^3+C*x^2+B*x+A)/(b*x+a)/(d*x+c)^(5/2),x)

[Out]

2*D*(d*x+c)^(1/2)/b/d^3-2/3/(a*d-b*c)/(d*x+c)^(3/2)*A+2/3/d/(a*d-b*c)/(d*x+c)^(3
/2)*B*c-2/3/d^2/(a*d-b*c)/(d*x+c)^(3/2)*C*c^2+2/3/d^3/(a*d-b*c)/(d*x+c)^(3/2)*D*
c^3+2/(a*d-b*c)^2/(d*x+c)^(1/2)*A*b-2/(a*d-b*c)^2/(d*x+c)^(1/2)*B*a+4/d/(a*d-b*c
)^2/(d*x+c)^(1/2)*C*a*c-2/d^2/(a*d-b*c)^2/(d*x+c)^(1/2)*C*b*c^2-6/d^2/(a*d-b*c)^
2/(d*x+c)^(1/2)*D*a*c^2+4/d^3/(a*d-b*c)^2/(d*x+c)^(1/2)*D*b*c^3+2*b^2/(a*d-b*c)^
2/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*A-2*b/(a*d-b*c
)^2/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*B*a+2/(a*d-b
*c)^2/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*C*a^2-2/b/
(a*d-b*c)^2/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*D*a^
3

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^3 + C*x^2 + B*x + A)/((b*x + a)*(d*x + c)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.239961, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^3 + C*x^2 + B*x + A)/((b*x + a)*(d*x + c)^(5/2)),x, algorithm="fricas")

[Out]

[1/3*(3*((D*a^3 - C*a^2*b + B*a*b^2 - A*b^3)*d^4*x + (D*a^3 - C*a^2*b + B*a*b^2
- A*b^3)*c*d^3)*sqrt(d*x + c)*log((sqrt(b^2*c - a*b*d)*(b*d*x + 2*b*c - a*d) + 2
*(b^2*c - a*b*d)*sqrt(d*x + c))/(b*x + a)) + 2*(8*D*b^2*c^4 - A*a*b*d^4 - 2*(7*D
*a*b + C*b^2)*c^3*d + (3*D*a^2 + 5*C*a*b - B*b^2)*c^2*d^2 - 2*(B*a*b - 2*A*b^2)*
c*d^3 + 3*(D*b^2*c^2*d^2 - 2*D*a*b*c*d^3 + D*a^2*d^4)*x^2 + 3*(4*D*b^2*c^3*d - (
7*D*a*b + C*b^2)*c^2*d^2 + 2*(D*a^2 + C*a*b)*c*d^3 - (B*a*b - A*b^2)*d^4)*x)*sqr
t(b^2*c - a*b*d))/((b^3*c^3*d^3 - 2*a*b^2*c^2*d^4 + a^2*b*c*d^5 + (b^3*c^2*d^4 -
2*a*b^2*c*d^5 + a^2*b*d^6)*x)*sqrt(b^2*c - a*b*d)*sqrt(d*x + c)), 2/3*(3*((D*a^
3 - C*a^2*b + B*a*b^2 - A*b^3)*d^4*x + (D*a^3 - C*a^2*b + B*a*b^2 - A*b^3)*c*d^3
)*sqrt(d*x + c)*arctan(-(b*c - a*d)/(sqrt(-b^2*c + a*b*d)*sqrt(d*x + c))) + (8*D
*b^2*c^4 - A*a*b*d^4 - 2*(7*D*a*b + C*b^2)*c^3*d + (3*D*a^2 + 5*C*a*b - B*b^2)*c
^2*d^2 - 2*(B*a*b - 2*A*b^2)*c*d^3 + 3*(D*b^2*c^2*d^2 - 2*D*a*b*c*d^3 + D*a^2*d^
4)*x^2 + 3*(4*D*b^2*c^3*d - (7*D*a*b + C*b^2)*c^2*d^2 + 2*(D*a^2 + C*a*b)*c*d^3
- (B*a*b - A*b^2)*d^4)*x)*sqrt(-b^2*c + a*b*d))/((b^3*c^3*d^3 - 2*a*b^2*c^2*d^4
+ a^2*b*c*d^5 + (b^3*c^2*d^4 - 2*a*b^2*c*d^5 + a^2*b*d^6)*x)*sqrt(-b^2*c + a*b*d
)*sqrt(d*x + c))]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x + C x^{2} + D x^{3}}{\left (a + b x\right ) \left (c + d x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x**3+C*x**2+B*x+A)/(b*x+a)/(d*x+c)**(5/2),x)

[Out]

Integral((A + B*x + C*x**2 + D*x**3)/((a + b*x)*(c + d*x)**(5/2)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.219336, size = 379, normalized size = 1.8 \[ -\frac{2 \,{\left (D a^{3} - C a^{2} b + B a b^{2} - A b^{3}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \sqrt{-b^{2} c + a b d}} + \frac{2 \,{\left (6 \,{\left (d x + c\right )} D b c^{3} - D b c^{4} - 9 \,{\left (d x + c\right )} D a c^{2} d - 3 \,{\left (d x + c\right )} C b c^{2} d + D a c^{3} d + C b c^{3} d + 6 \,{\left (d x + c\right )} C a c d^{2} - C a c^{2} d^{2} - B b c^{2} d^{2} - 3 \,{\left (d x + c\right )} B a d^{3} + 3 \,{\left (d x + c\right )} A b d^{3} + B a c d^{3} + A b c d^{3} - A a d^{4}\right )}}{3 \,{\left (b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}\right )}{\left (d x + c\right )}^{\frac{3}{2}}} + \frac{2 \, \sqrt{d x + c} D}{b d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^3 + C*x^2 + B*x + A)/((b*x + a)*(d*x + c)^(5/2)),x, algorithm="giac")

[Out]

-2*(D*a^3 - C*a^2*b + B*a*b^2 - A*b^3)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*
d))/((b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*sqrt(-b^2*c + a*b*d)) + 2/3*(6*(d*x + c
)*D*b*c^3 - D*b*c^4 - 9*(d*x + c)*D*a*c^2*d - 3*(d*x + c)*C*b*c^2*d + D*a*c^3*d
+ C*b*c^3*d + 6*(d*x + c)*C*a*c*d^2 - C*a*c^2*d^2 - B*b*c^2*d^2 - 3*(d*x + c)*B*
a*d^3 + 3*(d*x + c)*A*b*d^3 + B*a*c*d^3 + A*b*c*d^3 - A*a*d^4)/((b^2*c^2*d^3 - 2
*a*b*c*d^4 + a^2*d^5)*(d*x + c)^(3/2)) + 2*sqrt(d*x + c)*D/(b*d^3)